PhD Thesis

1. 1
Control relevant modeling and
nonlinear state estimation applied
to SOFC-GT power systems
Rambabu Kandepu
04-12-2007

2. 2
Contents
• Motivation
• Modeling and control of SOFC-GT
power system
• Nonlinear state estimation
• Conclusions

3. 3
Motivation
• Increase in energy demand
– Population growth
– Industrialization
• Dependency on oil and gas
• Global warming

4. 4
Motivation
• Solution to energy demand increase
– Efficient of energy conversion
– Technology with low emissions
– Using renewable energy sources
• Distributed generation
– Avoid transmission and distribution losses
– Wind turbines, biomass, small scale hydro, fuel cells etc

5. 5
Fuel cells
• Electrochemical device
• Advantages
– High efficiency
– Low emissions
– No moving parts
• Different types
– Electrolyte
– Temperature
• SOFC
– Solid components
– High operating temperature
– More fuel flexibility
– Internal reforming

6. 6
SOFC-GT system
Fuel Fuel cell
stack
Load
Gas
Air turbine
• Tight integration between SOFC and GT
• Low complexity models
– Relevant dynamics

7. 7
SOFC-GT system

8. 8
Modeling - SOFC
• Assumptions
– All variables are uniform
– Thermal inertia of gases is neglected
– Pressure losses are neglected for energy balance
– Ideal gas behavior
• Reactions
CH 4 + H 2O ⇔ CO + 3H 2
1 O + 2e − → O 2 −
2 2 CO + H 2O ⇔ CO2 + H 2
2− −
H2 + O → H 2 O + 2e CH 4 + 2 H 2O ⇔ CO2 + 4 H 2

9. 9
Modeling - SOFC
• Mass balance (anode and cathode)
dN i • • M
= N i ,in − N i ,out + ∑ aij rj
Anode
Electrolyte
dt j =1
Cathode
• Energy balance (one volume)
N N M
dTs
ms C s
P = − P + ∑ Fan ,i (han ,i − hi ) + ∑ Fca ,i (hca ,i − hi ) − ∑ ΔH j rj
dt i =1 i =1 j =1

10. 10
Modeling - SOFC
• Voltage
RT ⎛ pH 2 pO22 ⎞
1
E = E0 + ln ⎜ ⎟ V = E − Vloss
2 F ⎜ pH 2O
⎝
⎟
⎠
• Fuel Utilization (FU) = fuel utilized / fuel supplied
• Distributed nature of SOFC
• All models are developed in gPROMS
Fuel Anode inlet Anode outlet Anode inlet Anode outlet
Volume − I Volume − II
Air Cathode inlet Cathode outlet Cathode inlet Cathode outlet

11. 11
SOFC model evaluation
• Evaluated against a detailed model
1200
Detailed model
Simple model with one volume
1150 Simple model with two volumes
Temperature (K)
1100
1050
1000
950
0 100 200 300 400 500 600 700
Time (min)

12. 12
Control structure design
• Dynamic load operation is necessary
• Manipulated variable (1)
– Fuel flow rate
• Controlled variables (2)
– Fuel utilization (FU)
– SOFC temperature
• Load as a disturbance
• Need for a process redesign

13. 13
Control structure design
• Three possible options
– Air blow-off
– Extra fuel source
– Air by-pass
• Control structure
Load disturbance
FU ref
Fuel FU
Controller 1
flow
Tref Hybrid system T
Controller 2
Air blow-off
-

14. 14
SOFC-GT control

15. 15
SOFC-GT control
P
m fuel
FU
FUr PI FU
Controller 2
Hybrid System TSOFC
TSOFCr ωr
PI PI Ig ω
Controller 3 Controller 1
TSOFC ω

16. 16
SOFC-GT control – double shaft
Controlled variables
8
fuel flow rate (g/s)
6
4
2
0 5 10 15 20 25 30
time (sec)
Manipulated variables
air blow-off rate (kg/s)
0.1
0.05
0
0 5 10 15 20 25 30
time (sec)

17. 17
SOFC-GT control
• Model Predictive Control (MPC) to
include constraints
– FU
– Steam to carbon ratio
– SOFC temperature change
• Not all states are measurable
• State estimation is necessary

18. 18
State estimation
• Need for state estimation
• Nonlinear state estimation
– Extended Kalman Filter (EKF)
– Unscented Kalman Filter (UKF)
– Comparison
– Constraint handling
– Results
• Conclusions

19. 19
State estimation
• Important for process control and
performance monitoring
• Uncertainties; Model, measurement and
noise sources
• Represent the model state by an probability
distribution function (pdf)
• State estimation propagates the pdf over time
in some optimal way
• Gaussian pdf

20. 20
Nonlinear state estimation
• Extended Kalman Filter (EKF)
– Most common way to apply KF to a nonlinear system
• High order EKFs
– Computationally not feasible
• Ensemble Kalman Filter (EnKF)
– Mostly for large scale systems (reservoir models)
• Unscented Kalman Filter (UKF)
– Simple and effective
• Moving Horizon Estimation (MHE)
– Computationally demanding

21. 21
EKF principle
y = g ( x); x ∈ n
a random vector
g: n
→ m
, nonlinear function
(
How to compute the pdf of y, given the Gaussian pdf x, Px of x ?)
EKF
y = g ( x)
PyEKF = ( ∇g ) Px ( ∇g )
T
where ( ∇g ) is the Jacobian of g ( x) at x

22. 22
UKF principle
• UKF principle
y = g ( x); x ∈ n
a random vector
g: n
→ m
, nonlinear function
( )
How to compute the pdf of y, given the Gaussian pdf x, Px of x ?
UKF approximates the pdf.
It uses true nonlinear process and observation models.

23. 23
UKF principle
• UKF principle

24. 24
Comparison
• Example
= 58.26
= 2686

25. 25
EKF
Comparison UKF
110 110
Xmean
100 EKF
Ymean 100
Xmean
true
90 Ymean 90 ukf
Ymean
linearization
true
80 Px=16 80 Ymean
sigma points
true transformed sigma points
70 Py =2686 70
Px=16
EKF
60 Py =2304 60
y=g(x)=x2
y=g(x)=x2
true
Py =2686
50 50 UKF
Py =2816
40 40
30 30
20 20
10 10
0 0
0 5 10 0 5 10
x x
58.26

26. 26
Algorithms: EKF and UKF
Nonlinear system

27. 27
Algorithms: EKF and UKF
EKF UKF
Prediction step:
Calculate Jacobians /
sigma points
transformation
Prediction step:
Calculate mean and
covariance
Correction step:
Calculate Jacobians/
sigma points
transformation
Correction step:
Kalman update
equations

28. 28
State constraint handling
• No general way in KF theory
– Projecting unconstrained state estimate
onto boundary
• Systematic approach in MHE
– Solving a nonlinear problem at each time
step
• A simple method is introduced in UKF

29. 29
State constraint handling - EKF
xk−1
covariance
xkEKF, C
xkEKF

30. 30
State constraint handling - UKF
UKF, t=k
xk−1
Transformed sigma points
covariance
x-kUKF

31. 31
Constraint handling

32. 32
Constraint handling
UKF
• Constraint handling
method
– Projections at different steps
• Sigma points
• Transformed sigma points
• Transformed sigma points
through measurement
function
– Inequality constraints

33. 33
Constraint handling- example
• Gas phase reversible reaction
3
true
UKF
2 EKF
A
1
C
0
-1
0 1 2 3 4 5 6 7 8 9 10
time (sec)
true
4
UKF
EKF
3
B
C
2
1
0 1 2 3 4 5 6 7 8 9 10
time (sec)

34. 34
Comparison (EKF and UKF)
• Nonlinear systems
– Induction motor and Van der Pol Oscillator
– Faster convergence with UKF
• Robustness to model errors
– Van der Pol oscillator
• Better performance with UKF
• Higher order nonlinear system
– SOFC-GT hybrid system (18 states)

35. 35
Comparison (EKF and UKF)
Comparison of estimated states of an induction motor:
components of stator flux
1
true
UKF
EKF
0.5
1
x
0
0 5 10 15 20 25 30 35 40 45 50
time (sec)
0 true
UKF
EKF
-0.5
2
x
-1
-1.5
0 5 10 15 20 25 30 35 40 45 50
time (sec)

36. 36
Comparison (EKF and UKF)
• SOFC-GT system
– Higher order
nonlinear system
(18 states)
– Turbine shaft
speed plot

37. 37
Conclusions – state estimation
• The UKF is a promising option
– Simple and easy to implement
– No need for Jacobians
– Computational load is comparable to EKF
– Improved performance
• Faster convergence
• Robustness to model errors and initial choices
• Simple constraint handling method works

38. 38
Thank you
for
your
attention
☺